We identify two objectives to learn a model \(h\) , expected risk minimization (ERM) and group minimax fairness (GMMF), a relaxed version of [1]}, [2]}, We note that they share the objective with group distributional robustness [3]}. which is already robust to group shifts. We present their statistical formulations, and refer the reader to Appendix REF  for implementation details,
\(\begin{array}{rl}\text{(ERM):}\; \min \limits _{h \in \mathcal {H}} \mathbb {E}_{P}[\ell (h(X),Y)],& \text{(GMMF):}\; \min \limits _{h \in \mathcal {H}} \max \limits _{\begin{array}{c}Q_G \ge \epsilon \end{array}}\mathbb {E}_{P}[\frac{Q_G}{P_G}\ell (h(X),Y)],\end{array}\) 
